The Finite Simple Groups (Graduate Texts in Mathematics) 2009 edition by Wilson, Robert (2009) Hardcover

Graduate Texts in Mathematics #251

by Robert A. Wilson

The finite simple groups are the building blocks from which all the finite groups are made and as such they are objects of fundamental importance throughout mathematics. The classification of the finite simple groups was one of the great mathematical achievements of the twentieth century, yet these groups remain difficult to study which hinders applications of the classification. This textbook brings the finite simple groups to life by giving concrete constructions of most of them, sufficient to illuminate their structure and permit real calculations both in the groups themselves and in the underlying geometrical or algebraic structures. This is the first time that all the finite simple groups have been treated together in this way and the book points out their connections, for example between exceptional behaviour of generic groups and the existence of sporadic groups, and discusses a number of new approaches to some of the groups. Many exercises of varying difficulty are provided.

Genres: Mathematics

Hardcover

First published January 1, 2009

About the author

Robert A. Wilson was proprietor of the famous Phoenix Bookshop in New York City. He was a passionate writer and author of bibliographies of Gertrude Stein, Gregory Corso, and Denise Levertov. Wilson specialized in rare books and manuscripts.

Reviews

[Steve Dalton · Rating 2 out of 5]

This is one of those books that you just kind of race to finish; a book that you just want to get through. There were times where my eyes glazed over in the face of pages of tedious calculation, with no insight whatsoever into what was going on. In fact, that happened a lot.

And I get the feeling it's because the book is sort of written that way. This feels very much like a rushed rough draft: there are times when Wilson is discussing a group for pages on end, never once mentioning the group is the automorphism group of the particular structure he is meticulously detailing. Several times I had to Google to see if, in fact, yes, the sporadic Suzuki group is the automorphism group of the complex Leech lattice.

I actually had planned to give this book one star, but there are some saving graces. The first chapter on alternating groups is well-written, and provides nice insight into the kind of analysis one can expect in classifying finite simple groups. The last chapter on sporadic simple groups, while not nearly as well-written, is incredibly detailed in the construction of the Golay code and the Leech lattice. The later sections on the Fischer groups, the first generation of the Happy Family (which includes the Monster), and the Pariahs, are all hodge-podge without much to say. One could argue that not much is known as far as a unified approach to these groups goes. But nothing is said about the history, initial constructions, etc. of these groups either. In fact, if this was your first foray into the world of finite simple groups, you would get the impression most group theorists spend their time constructing esoteric graphs and computing their automorphisms.

So while this book offers some insight, it completely spurns the insight of the numerous group theorists who first conjectured - then proved - the existence of many of these groups. Why did Wilson not include any information on this, save the few parenthetical asides? When reading about how the Monster is constructed, for example, why not include how some conjectured its existence before it was ever constructed, or how anyone ever got the idea of looking at graph automorphisms to construct groups? It just seems pulled out of thin air.

I cannot recommend this book to anyone interested in Finite Simple Groups. There are better overviews out there, at a more introductory level. If you're interested in a particular class of FSG, either browse Wilson's extensive bibliography (and see his Further Reading sections), or simply Google the groups you want to know more about, and be presented with a plethora of nice results. Indeed, if you choose to read this book, be prepared to use Google quite often, to either fill in details that are skipped, or to provide alternate methods of construction when Wilson gets jammed up on details. I might be tempted to recommend this book solely for the great treatment of the Golay code and the Leech lattice, but even Wilson admits Griess wrote a great book on the associated Conway groups [Twelve Sporadic Groups]. If, against my advice, you read this book, please don't get bogged down in details: the author's already done that for you.

[Alex Nelson · Rating 5 out of 5]

This is the best introduction to finite simple groups, period.

The author provides insight behind most of the groups (the symmetric group should be "self-motivating").

He notes many different constructions, and provides citations to wonderful introductions...so although it's not a "100% self-contained book", it is nevertheless the best. (And the author provides the simplest, clearest constructions of each of the finite simple groups --- only the pariahs are a bit random, because they are exceptions!)

Each chapter ends with about 100 exercises. They're quite difficult, and I'm still trying to figure out what some teach.

I'd highly recommend this book for neophytes to finite simple group theory.